1. S. Grivopoulos. Optimal Control of Quantum Systems. PhD thesis, ME, University of California at Santa Barbara, December 2005. Abstract:

   Motivated by developments and problems in a number of disciplines including Quantum Chemistry, Information and Optics, the theory of Control of Quantum Systems has emerged. Its goal is to apply the tools and methods of Control Theory in the analysis and design of scientific and engineering applications of Quantum Systems. At the same time, Control Theory itself is been enriched by new models and paradigms. Our work focuses on the design of control fields that achieve given state transfers with the minimum amount of energy expenditure. Besides their inherent mathematical interest, such optimal designs are closely tied to the dynamics of the underlying system and reveal much about the interplay of dynamics and control. In the first part of our work, we consider energy-optimal transfers in a general isolated quantum system, for example an atom or a molecule. By examining the large-time limit of these optimal transfer problems, we uncover the general structure of the optimal controls. Moreover, we reduce the computational complexity of the problem significantly. In the second part, we examine similar problems for open quantum systems, that is, quantum systems that interact with their environment. This interaction creates dissipative effects in the system. Although one usually wants to resist these effects, there are instances, such as the cooling of internal molecular motion, that one can effectively use dissipation mechanisms to one's advantage. We apply techniques similar to those developed for isolated systems to design ``cooling'' electric (laser) fields for molecular rotations and demonstrate our method in a three-state Lambda system. The aforementioned designs are open-loop because sensing and feedback are infeasible for the applications at hand. However, there has been remarkable technological progress in the monitoring of single/small numbers of quantum systems. Also, from the theoretical side, an adequate formalism has been developed for the description of continuously monitored quantum systems along with a theory of optimal feedback. In the third part, we capitalize on these developments and design an optimal feedback control for the stabilization/preparation of a desired state of a continuously measured spin system.

   
   

   @PhdThesis{phd-05-sg,
   author = "S. Grivopoulos ",
   title = "Optimal Control of Quantum Systems",
   school = {ME, University of California at Santa Barbara},
   year = 2005,
   month = dec,
   pdf = http://ccdc.mee.ucsb.edu/pdf/phd-05-sg.pdf,
   abstract = { Motivated by developments and problems in a number of disciplines including Quantum Chemistry, Information and Optics, the theory of Control of Quantum Systems has emerged. Its goal is to apply the tools and methods of Control Theory in the analysis and design of scientific and engineering applications of Quantum Systems. At the same time, Control Theory itself is been enriched by new models and paradigms. Our work focuses on the design of control fields that achieve given state transfers with the minimum amount of energy expenditure. Besides their inherent mathematical interest, such optimal designs are closely tied to the dynamics of the underlying system and reveal much about the interplay of dynamics and control. In the first part of our work, we consider energy-optimal transfers in a general isolated quantum system, for example an atom or a molecule. By examining the large-time limit of these optimal transfer problems, we uncover the general structure of the optimal controls. Moreover, we reduce the computational complexity of the problem significantly. In the second part, we examine similar problems for open quantum systems, that is, quantum systems that interact with their environment. This interaction creates dissipative effects in the system. Although one usually wants to resist these effects, there are instances, such as the cooling of internal molecular motion, that one can effectively use dissipation mechanisms to one's advantage. We apply techniques similar to those developed for isolated systems to design ``cooling'' electric (laser) fields for molecular rotations and demonstrate our method in a three-state Lambda system. The aforementioned designs are open-loop because sensing and feedback are infeasible for the applications at hand. However, there has been remarkable technological progress in the monitoring of single/small numbers of quantum systems. Also, from the theoretical side, an adequate formalism has been developed for the description of continuously monitored quantum systems along with a theory of optimal feedback. In the third part, we capitalize on these developments and design an optimal feedback control for the stabilization/preparation of a desired state of a continuously measured spin system. } 
   }
   




2. R. Skjetne. The Maneuvering Problem. PhD thesis, Norwegian University of Science and Technology, Dept. Eng. Cybernetics, Trondheim, Norway, March 2005.
   

   @PHDTHESIS{phd-05-rs,
   AUTHOR = {R. Skjetne},
   TITLE = {The Maneuvering Problem},
   SCHOOL = {Norwegian University of Science and Technology},
   pdf = http://ccdc.mee.ucsb.edu/pdf/phd-05-rs.pdf,
   YEAR = 2005,
   month = mar,
   ADDRESS = {Dept. Eng. Cybernetics, Trondheim, Norway},
   
   }
   




3. S. E. Tuna. Generalized Dilations and Homogeneity. PhD thesis, University of California at Santa Barbara, 2005. Abstract:

   Within systems theory homogeneity was hitherto defined with respect to a class of operators named dilations. We observe that for discrete-time systems it is natural to define homogeneity with respect to a more general class which we name generalized dilations. Considering homogeneity in this broader sense we devise a numerical algorithm to obtain optimization-based, offline-computed, globally stabilizing feedback laws for discrete-time homogeneous systems. Our second observation is that a continuous-time homogeneous (with respect to a dilation) system yields a homogeneous discrete-time model if one adopts a sample and hold strategy where the hold intervals depend on the state. Combination of this fact with the numerical algorithm developed for discrete-time homogeneous systems yields a practical method to robustly stabilize continuous-time homogeneous (with respect to a dilation) systems. We show that the method can be used in robust global exponential stabilization of chained systems and systems in power form. Thirdly, for continuous-time systems we introduce homogeneity with respect to generalized dilations in terms of the system trajectories. We then obtain the condition that homogeneity implies on the righthand side of the differential equation describing the system. We observe that any system with unique solutions in forward and backward time is homogeneous with respect to some generalized dilation. We also generate results with insights into concepts such as monotonicity and finite-time convergence under homogeneity. We finally study discrete-time homogeneous (with respect to a dilation) systems under arbitrary switching. We propose an optimization-based, constructive method to generate a homogeneous control Lyapunov function and an associated feedback law that guarantees robust asymptotic stability for all possible switching scenarios. We observe that for linear systems the resulting Lyapunov function turns out to be convex.

   
   

   @PhdThesis{phd-05-set,
   author = "S. E. Tuna",
   title = "Generalized Dilations and Homogeneity",
   school = "University of California at Santa Barbara",
   pdf = http://ccdc.mee.ucsb.edu/pdf/phd-05-set.pdf,
   year = 2005,
   abstract = {Within systems theory homogeneity was hitherto defined with respect to a class of operators named dilations. We observe that for discrete-time systems it is natural to define homogeneity with respect to a more general class which we name generalized dilations. Considering homogeneity in this broader sense we devise a numerical algorithm to obtain optimization-based, offline-computed, globally stabilizing feedback laws for discrete-time homogeneous systems. Our second observation is that a continuous-time homogeneous (with respect to a dilation) system yields a homogeneous discrete-time model if one adopts a sample and hold strategy where the hold intervals depend on the state. Combination of this fact with the numerical algorithm developed for discrete-time homogeneous systems yields a practical method to robustly stabilize continuous-time homogeneous (with respect to a dilation) systems. We show that the method can be used in robust global exponential stabilization of chained systems and systems in power form. Thirdly, for continuous-time systems we introduce homogeneity with respect to generalized dilations in terms of the system trajectories. We then obtain the condition that homogeneity implies on the righthand side of the differential equation describing the system. We observe that any system with unique solutions in forward and backward time is homogeneous with respect to some generalized dilation. We also generate results with insights into concepts such as monotonicity and finite-time convergence under homogeneity. We finally study discrete-time homogeneous (with respect to a dilation) systems under arbitrary switching. We propose an optimization-based, constructive method to generate a homogeneous control Lyapunov function and an associated feedback law that guarantees robust asymptotic stability for all possible switching scenarios. We observe that for linear systems the resulting Lyapunov function turns out to be convex.} 
   }
   

1. M. Fardad and B. Bamieh. On Stability and the Spectrum Determined Growth Condition for Spatially Periodic Systems. Technical report CCDC-05-1125, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005.
   

   @TechReport{CCDC-05-1125,
   author = {M. Fardad and B. Bamieh},
   title = {On Stability and the Spectrum Determined Growth Condition for Spatially Periodic Systems},
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   year = 2005,
   number = {CCDC-05-1125},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-1125.pdf 
   }
   




2. M. Fardad and B. Bamieh. The Nyquist Stability Criterion For A Class Of Spatially Periodic Systems. Technical report CCDC-05-1124, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005.
   

   @TechReport{CCDC-05-1124,
   author = {M. Fardad and B. Bamieh},
   title = {The Nyquist Stability Criterion For A Class Of Spatially Periodic Systems},
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   year = 2005,
   number = {CCDC-05-1124},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-1124.pdf 
   }
   




3. M. Fardad, M. R. Jovanovic, and B. Bamieh. Frequency Analysis and Norms of Distributed Spatially Periodic Systems. Technical report CCDC-05-1126, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005.
   

   @TechReport{CCDC-05-1126,
   author = {M. Fardad and M. R. Jovanovi\'c and B. Bamieh},
   title = {Frequency Analysis and Norms of Distributed Spatially Periodic Systems},
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   year = 2005,
   number = {CCDC-05-1126},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-1126.pdf 
   }
   




4. M. Fardad, M. R. Jovanovic, and B. Bamieh. Stability, Norms and Frequency Analysis of Distributed Spatially Periodic Systems. Technical report CCDC-05-0720, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005.
   

   @TechReport{CCDC-05-0720,
   author = {M. Fardad and M. R. Jovanovi\'c and B. Bamieh},
   title = {Stability, Norms and Frequency Analysis of Distributed Spatially Periodic Systems},
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   year = 2005,
   number = {CCDC-05-0720},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-0720.pdf,
   
   }
   




5. B. Munsky and M. Khammash. The Finite State Projection Algorithm for the Solution of the Chemical Master Equation. Technical report CCDC-05-0505, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005. Abstract:

   This article introduces the Finite State Projection (FSP) method for use in the stochastic analysis of chemically reacting systems. The chemical population of such systems have probability density vectors that evolve according to a set of ordinary differential equations known as the Chemical Master Equation (CME). If the CME describes a system that has only a finite number of distinct population vectors, the FSP method provides an exact analytical solution to the CME. When an infinite or extremely large number of population variations are possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how well the solution on the truncated space approximates the full system. A FSP algorithm is developed that systematically increases the projection space in order to meet pre-specified tolerance in the probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP and Stochastic Simulation Algorithm (SSA) results. In both examples, the FSP vastly outperforms the SSA in terms of accuracy as well as computational efficiency.

   
   

   @TechReport{CCDC-05-0505,
   AUTHOR = {B. Munsky and M. Khammash},
   TITLE = {The Finite State Projection Algorithm for the Solution of the Chemical Master Equation},
   number = {CCDC-05-0505},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-0505.pdf,
   year = 2005,
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   abstract = {This article introduces the Finite State Projection (FSP) method for use in the stochastic analysis of chemically reacting systems. The chemical population of such systems have probability density vectors that evolve according to a set of ordinary differential equations known as the Chemical Master Equation (CME). If the CME describes a system that has only a finite number of distinct population vectors, the FSP method provides an exact analytical solution to the CME. When an infinite or extremely large number of population variations are possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how well the solution on the truncated space approximates the full system. A FSP algorithm is developed that systematically increases the projection space in order to meet pre-specified tolerance in the probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP and Stochastic Simulation Algorithm (SSA) results. In both examples, the FSP vastly outperforms the SSA in terms of accuracy as well as computational efficiency. },
   
   }
   




6. O. Storset and B. Paden. Discrete track electrodynamic maglev. Part I: Modelling. Technical report CCDC-05-0930, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005.
   

   @TechReport{CCDC-05-0930,
   author = {O. Storset and B. Paden},
   title = {Discrete track electrodynamic maglev. {P}art {I}: {M}odelling},
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   year = 2005,
   number = {CCDC-05-0930},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-0930.pdf,
   
   }
   




7. O. Storset and B. Paden. Electrodynamic magnetic levitation with discrete track. Part II: Periodic track model for numerical simulation and lumped parameter model. Technical report CCDC-05-1001, Center for Control Engineering and Computation. University of California at Santa Barbara, 2005.
   

   @TechReport{CCDC-05-1001,
   author = {O. Storset and B. Paden},
   title = {Electrodynamic magnetic levitation with discrete track. {Part II:} {P}eriodic track model for numerical simulation and lumped parameter model},
   institution = {Center for Control Engineering and Computation. University of California at Santa Barbara},
   year = 2005,
   number = {CCDC-05-1001},
   pdf = http://ccdc.mee.ucsb.edu/pdf/ccdc-05-1001.pdf,
   
   }
   

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